Closed surfaces with bounds on their Willmore energy
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 3, p. 605-634

The Willmore energy of a closed surface in ${ℝ}^{n}$ is the integral of its squared mean curvature, and is invariant under Möbius transformations of ${ℝ}^{n}$. We show that any torus in ${ℝ}^{3}$ with energy at most $8\pi -\delta$ has a representative under the Möbius action for which the induced metric and a conformal metric of constant (zero) curvature are uniformly equivalent, with constants depending only on $\delta >0$. An analogous estimate is also obtained for closed, orientable surfaces of fixed genus $p\ge 1$ in ${ℝ}^{3}$ or ${ℝ}^{4}$, assuming suitable energy bounds which are sharp for $n=3$. Moreover, the conformal type is controlled in terms of the energy bounds.

Published online : 2019-02-22
Classification:  53A05,  53A30,  53C21,  49Q15
@article{ASNSP_2012_5_11_3_605_0,
author = {Kuwert, Ernst and Sch\"atzle, Reiner},
title = {Closed surfaces with bounds on their Willmore energy},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 11},
number = {3},
year = {2012},
pages = {605-634},
zbl = {1260.53027},
mrnumber = {3059839},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2012_5_11_3_605_0}
}

Kuwert, Ernst; Schätzle, Reiner. Closed surfaces with bounds on their Willmore energy. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 3, pp. 605-634. http://www.numdam.org/item/ASNSP_2012_5_11_3_605_0/

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